Wittgenstein on Mathematical Proof

1990 ◽  
Vol 28 ◽  
pp. 79-99 ◽  
Author(s):  
Crispin Wright
Keyword(s):  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Short Paper ◽  
Foundations Of Mathematics ◽  
The Real ◽  
Philosophical Investigations

To be asked to provide a short paper on Wittgenstein's views on mathematical proof is to be given a tall order (especially if little or no familiarity either with mathematics or with Wittgenstein's philosophy is to be presupposed!). Close to one half of Wittgenstein's writings after 1929 concerned mathematics, and the roots of his discussions, which contain a bewildering variety of underdeveloped and sometimes conflicting suggestions, go deep to some of the most basic and difficult ideas in his later philosophy. So my aims in what follows are forced to be modest. I shall sketch an intuitively attractive philosophy of mathematics and illustrate Wittgenstein's opposition to it. I shall explain why, contrary to what is often supposed, that opposition cannot be fully satisfactorily explained by tracing it back to the discussions of following a rule in the Philosophical Investigations and Remarks on the Foundations of Mathematics. Finally, I shall try to indicate very briefly something of the real motivation for Wittgenstein's more strikingly deflationary suggestions about mathematical proof, and canvass a reason why it may not in the end be possible to uphold them.

scholarly journals Rules, understanding and language games in mathematics

2021 ◽  
pp. 35-45
Author(s):  
V. V. Tselishchev
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Language Game ◽  
Language Games ◽  
Logical Necessity ◽  
Foundations Of Mathematics ◽  
Mathematical Concepts ◽  
Philosophical Investigations ◽  
Form Of Life ◽  
Inherent Ambiguity

The article is devoted to the applicability of Wittgenstein’s following the rule in the context of his philosophy of mathematics to real mathematical practice. It is noted that in «Philosophical Investigations» and «Remarks on the Foundations of Mathematics» Wittgenstein resorted to the analysis of rather elementary mathematical concepts, accompanied also by the inherent ambiguity and ambiguity of his presentation. In particular, against this background, his radical conventionalism, the substitution of logical necessity with the «form of life» of the community, as well as the inadequacy of the representation of arithmetic rules by a language game are criticized. It is shown that the reconstruction of the Wittgenstein concept of understanding based on the Fregian division of meaning and referent goes beyond the conceptual framework of Wittgenstein language games.

Platonism and Mathematical Intuition in Kurt Gödel's Thought

1995 ◽  
Vol 1 (1) ◽  
pp. 44-74 ◽  
Author(s):  
Charles Parsons
Keyword(s):  
Set Theory ◽  
Real World ◽  
Mathematical Knowledge ◽  
Philosophy Of Mathematics ◽  
Foundations Of Mathematics ◽  
Mathematical Objects ◽  
The Real ◽  
Robust Realism ◽  
Mathematical Intuition

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish.

Coetzee’s Quest for Reality

2018 ◽  
Author(s):  
Alice Crary
Keyword(s):  
Philosophical Discourse ◽  
The Real ◽  
Philosophical Investigations ◽  
Literature And Philosophy ◽  
Universal Truth

In this chapter, Alice Crary argues that a truly ‘realist’ work of literature might be one that, instead of conforming to familiar genre-specifications, attempts by other means to expose readers to the real—that is, to how things really are. Crary highlights Coetzee’s efforts to elicit what she calls ‘transformative thought’: a process that involves both delineating the progress of individual characters in their quests for reality, and, in formal terms, inviting readers to, for instance, imaginatively participate in such quests. With regard to The Childhood of Jesus, she highlights resonances between these features of Coetzee’s writing and Wittgenstein’s procedures in the Philosophical Investigations. In doing so, Crary brings out a respect in which literature and philosophy are complementary discourses: literature can deal in the sort of objective or universal truth that is philosophy’s touchstone, and philosophical discourse can have an essentially literary dimension.

A Markov Prediction-Based Privacy Protection Scheme for Continuous Query

2019 ◽  
Vol 28 (09) ◽  
pp. 1950147
Author(s):  
Lei Zhang ◽  
Jing Li ◽  
Songtao Yang ◽  
Yi Liu ◽  
Xu Zhang ◽  
...  
Keyword(s):  
Privacy Protection ◽  
Location Privacy ◽  
Mathematical Proof ◽  
Security Analysis ◽  
Continuous Query ◽  
The Real ◽  
Protection Scheme ◽  
Whole Process ◽  
Prediction Scheme ◽  
Location Privacy Protection

The query probability of a location which the user utilizes to request location-based service (LBS) can be used as background knowledge to infer the real location, and then the adversary may invade the privacy of this user. In order to cope with this type of attack, several algorithms had provided query probability anonymity for location privacy protection. However, these algorithms are all efficient just for snapshot query, and simply applying them in the continuous query may bring hazards. Especially that, continuous anonymous locations which provide query probability anonymity in continuous anonymity are incapable of being linked into anonymous trajectories, and then the adversary can identify the real trajectory as well as the real location of each query. In this paper, the query probability anonymity and anonymous locations linkable are considered simultaneously, then based on the Markov prediction, we provide an anonymous location prediction scheme. This scheme can cope with the shortage of the existing algorithms of query probability anonymity in continuous anonymity locations difficult to be linked, and provide query probability anonymity service for the whole process of continuous query, so this scheme can be used to resist the attack of both of statistical attack as well as the infer attack of the linkable. At last, in order to demonstrate the capability of privacy protection in continuous query and the efficiency of algorithm execution, this paper utilizes the security analysis and experimental evaluation to further confirm the performance, and then the process of mathematical proof as well as experimental results are shown.

Paul Benacerraf and Hilary Putnam. Introduction. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, 1964, pp. 1–27. - Rudolf Carnap. The logicist foundations of mathematics. English translation of 3528 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 31–41. - Arend Heyting. The intuitionist foundations of mathematics. English translation of 3856 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 42–49. - Johann von Neumann. The formalist foundations of mathematics. English translation of 2998 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 50–54. - Arend Heyting. Disputation. A reprint of pages 1-12 (the first chapter) and parts of the bibliography of XXI 367. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 55–65. - L. E. J. Brouwer. Intuitionism and formalism. A reprint of 1557. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 66–77. - L. E. J. Brouwer. Consciousness, philosophy, and mathematics. A reprint of pages 1243-1249 of XIV 132. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 78–84. - Gottlob Frege. The concept of number. English translation of pages 67-104, 115-119, of 495 (1884 edn.) by Michael S. Mahoney. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 85–112. - Bertrand Russell. Selections from Introduction to mathematical philosophy. A reprint of pages 1-19, 194-206, of 11126 (1st edn., 1919). Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 113–133. - David Hilbert. On the infinite. English translation of 10813 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 134–151.

1969 ◽  
Vol 34 (1) ◽  
pp. 107-110
Author(s):  
Alec Fisher
Keyword(s):  
New Jersey ◽  
English Translation ◽  
Philosophy Of Mathematics ◽  
Bertrand Russell ◽  
Hilary Putnam ◽  
Foundations Of Mathematics ◽  
Rudolf Carnap ◽  
Von Neumann ◽  
David Hilbert ◽  
And Mathematics

Wittgenstein on Representation, Privileged Objects, and Private Languages

1983 ◽  
Vol 13 (1) ◽  
pp. 57-78 ◽  
Author(s):  
Meredith Williams
Keyword(s):  
Good Reason ◽  
Private Language ◽  
The Real ◽  
Philosophical Investigations ◽  
Philosophical Significance ◽  
Private Language Argument

In this paper, I shall investigate Wittgenstein's ‘private language argument,’ that is, the argument to be found in Philosophical Investigations 243-315. Roughly, this argument is intended to show that a language knowable to one person and only that person is impossible; in other words, a ‘language’ which another person cannot understand isn't a language. Given the prolonged debate sparked by these passages, one must have good reason to bring it up again. I have: Wittgenstein's attack on private languages has regularly been misinterpreted. Moreover, it has been misinterpreted in a way that draws attention away from the real force of his arguments and so undercuts the philosophical significance of these passages.

Brouwer versus Hilbert: 1907–1928

1998 ◽  
Vol 11 (2) ◽  
pp. 291-325 ◽  
Author(s):  
J. Posy Carl
Keyword(s):  
Twentieth Century ◽  
Current Literature ◽  
Philosophy Of Mathematics ◽  
Formal Logic ◽  
Foundations Of Mathematics ◽  
Theory Of Meaning ◽  
Classical Mathematics ◽  
And Behavior ◽  
Shed Light ◽  
The Way

The ArgumentL. E. J. Brouwer and David Hubert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notorious Grundlagenstreit centered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his “Intuitionism” did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics.Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, misconstruals which last to our own time: Current literature often portrays popular views of intuitionism as the product of Brouwer's idiosyncratic subjectivism; modern logicians view intuitionism as simply applying a non-standard formal logic to mathematics; and contemporary philosophers see that logic as based upon a pure assertabilist theory of meaning. These pictures stem from the way Hilbert structured the controversy.Even though Brouwer's own work and behavior occasionally reinforce these pictures, they are nevertheless inaccurate accounts of his approach to mathematics. However, the framework provided by the Brouwer-Hilbert debate itself does not supply an adequate correction of these inaccuracies. For, even if we eliminate these mistakes within that framework, Brouwer's position would still appear fragmented and internally inconsistent. I propose a Kantian framework — not from Kant's philosophy of mathematics but from his general metaphysics — which does show the coherence and consistency of Brouwer's views. I also suggest that expanding the context of the controversy in this way will illuminate Hilbert's views as well and will even shed light upon Kant's philosophy.

scholarly journals Proof vs Truth in Mathematics

Studia Humana ◽  
2020 ◽  
Vol 9 (3-4) ◽  
pp. 10-18
Author(s):  
Roman Murawski
Keyword(s):  
Philosophy Of Mathematics ◽  
Foundations Of Mathematics ◽  
Research Practice ◽  
Formal Proofs ◽  
Concept Of Truth

AbstractTwo crucial concepts of the methodology and philosophy of mathematics are considered: proof and truth. We distinguish between informal proofs constructed by mathematicians in their research practice and formal proofs as defined in the foundations of mathematics (in metamathematics). Their role, features and interconnections are discussed. They are confronted with the concept of truth in mathematics. Relations between proofs and truth are analysed.

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